Question

Suppose that two identical firms produce widgets and that they are the only firms in the market. Their costs are given by C1 = 60Q1 and C2 = 60Q2, where Q1 is the output of Firm 1 and Q2 the output of Firm 2. Price is determined by the following demand curve P = 300 – Q where Q = Q1 + Q2.

1. Find the Cournot-Nash equilibrium. Calculate the profit of each firm at this equilibrium.
2. Suppose the two firms form a cartel to maximize joint profits. How many widgets will be produced? Calculate each firm’s profit.
3. Suppose Firm 1 were the only firm in the industry. How would market output and Firm 1’s profit differ from that found in part (b) above?
4. Returning to the duopoly of part (b), suppose Firm 1 abides by the agreement, but Firm 2 cheats by increasing production. How many widgets will Firm 2 produce? What will be each firm’s profits?

Short answer

1. Cournot Nash equilibrium is 80 units of output and $140 price. Profit for each firm will be $6,400.
2. The cartel will produce 120 widgets at $180. The profit of each firm will be $7.200.
3. The market output will be 120 widgets, and the profit will be $14,400.
4. Firm 2 will produce 90 widgets. Firm 1 profit will be $5,400, and firm 2 profit will be $8,100.

Step-by-step solution

Explanation for part (a)

The Cournot Nash equilibrium occurs where two firms decide the output level simultaneously.

The reaction curve of firm 1 is calculated below:

$$\begin{gathered} P=300-Q_1-Q_2 \\ {C}_1=60Q_1 \\ {\mathrm{\pi }}_1={\mathrm{TR}}_1-{\mathrm{C}}_1 \\ ={\mathrm{PQ}}_1-{\mathrm{C}}_1 \\ =300{\mathrm{Q}}_1-{\mathrm{Q}}_1^2-{\mathrm{Q}}_1{\mathrm{Q}}_2-60{\mathrm{Q}}_1 \\ \frac{{\mathrm{d\pi }}_1}{{\mathrm{dQ}}_1}=300-2{\mathrm{Q}}_1-{\mathrm{Q}}_2-60=0 \\ 240-2{\mathrm{Q}}_1-{\mathrm{Q}}_2=0 \\ {\mathrm{Q}}_1=\frac{240-{\mathrm{Q}}_2}{2}...................\mathrm{i} \end{gathered}$$

The reaction curve for firm 2 is calculated below:

$$\begin{gathered} {\mathrm{\pi }}_2={\mathrm{TR}}_2-{\mathrm{C}}_2 \\ ={\mathrm{PQ}}_2-{\mathrm{C}}_2 \\ =300{\mathrm{Q}}_2-{\mathrm{Q}}_2^2-{\mathrm{Q}}_1{\mathrm{Q}}_2-60{\mathrm{Q}}_2 \\ \frac{{\mathrm{d\pi }}_2}{\mathrm{dQ}2}=300-2{\mathrm{Q}}_2-{\mathrm{Q}}_1-60=0 \\ 240-2{\mathrm{Q}}_2-{\mathrm{Q}}_1=0 \\ {\mathrm{Q}}_2=\frac{240-{\mathrm{Q}}_1}{2}...................\mathrm{ii} \end{gathered}$$

From i and ii,

$$\begin{gathered} Q_1=\frac{240-\left(\frac{240-\mathrm{Q}1}{2}\right)}{2} \\ =\frac{240+{\mathrm{Q}}_1}{4} \\ 4Q_1-Q_1=240 \\ {Q}_1=\frac{240}{3}=80 \\ {Q}_2=\frac{240-80}{2}=80 \end{gathered}$$

The market price is calculated below:

$$\begin{gathered} Q=80+80 \\ =160 \\ P=300-160 \\ =\$140 \end{gathered}$$

The profit for each firm is calculated below:

$$\begin{gathered} {\mathrm{\pi }}_1=\left(140\times 80\right)-\left(60\times 80\right) \\ =11,200-4,800 \\ =\$6,400 \\ {\mathrm{\pi }}_2=\left(140\times 80\right)-\left(60\times 80\right) \\ =11,200-4,800 \\ =\$6,400 \end{gathered}$$

Thus, the profit of firm 1 and firm 2 will be $6,400.

Explanation for part (b)

The cartel maximizes the industry profit, thus fixing the output level.

The price and output of the cartel are calculated below:

$$\begin{gathered} P=300-Q \\ TR=300Q-Q^2 \\ MR=300-2Q \\ C=60Q \\ MC=60 \\ MR=MC \\ 300-2Q=60 \\ 2Q=240 \\ Q=120 \\ P=300-120 \\ =\$180 \end{gathered}$$

The cartel output will be 120 widgets, and the price will be $180. The cartel output will be divided among the two firms; thus, each firm will produce 60 units.

The profit will be the same for both firms as the marginal cost is the same for both firms. The profit is calculated below:

$$\begin{gathered} \mathrm{\pi }=\left(180\times 60\right)-\left(60\times 60\right) \\ =10,800-3600 \\ =\$7,200 \end{gathered}$$

The profit of each firm will be $7,200.

Explanation for part (c)

If firm 1 is the only firm in the industry, then the entire output, i.e., 120 widgets, will be produced by firm 1 at $180. The profit of firm 1 is calculated below:

$$\begin{gathered} \mathrm{\pi }=\left(180\times 120\right)-\left(60\times 120\right) \\ =21,600-7,200 \\ =\$14,400 \end{gathered}$$

The profit of firm 1 will be $14,400.

Explanation for part (d)

Assume that the output is split equally between the firms, then firm 1 and firm 2 produce 60 units each. The output of firm 2 can be calculated from firm 2’s reaction curve and the output of firm 1.

The output of firm 2 when it cheats the agreement is calculated below:

$$\begin{gathered} Q_1=60 \\ {Q}_2=\frac{240-60}{2} \\ =\frac{180}{2} \\ =90 \end{gathered}$$

The price is calculated below:

$$\begin{gathered} Q_T=Q_1+Q_2 \\ =90+60 \\ =150 \\ P=300-150 \\ =\$150 \end{gathered}$$

The profit is each firm is calculated below:

$$\begin{gathered} {\mathrm{\pi }}_1=\left(150\times 60\right)-\left(60\times 60\right) \\ =9000-3600 \\ =\$5400 \\ {\mathrm{\pi }}_2=\left(150\times 90\right)-\left(60\times 90\right) \\ =13,500-5400 \\ =\$8,100 \end{gathered}$$

Firm 2 earns a higher profit by cheating the agreement.